The arithmetic mean (average) M of 4 terms is an integer.

Question

The arithmetic mean (average) M of 4 terms is an integer. When M is divided by 16, the remainder is 10. If each of the terms is increased by 100%, what is the remainder when the new mean is divided by 16?

A. 1
B. 4
C. 10
D. 12
E. 14

A. 1
B. 4
C. 10
D. 12
E. 14

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MarkGMATMentor · Answer

This is both a quotient / remainder problem and an average problem, and it has good takeaways for both aspects.

On remainder problems with an unknown total, picking numbers is often a good strategy. And on average problems, when you apply the same operation to every member of a set (adding, subtracting, multiplying, or dividing by the same number), you can apply the same operation to the average itself.

First, note that you can pick a number for M.  Since the answer choices are all numbers, as long as you pick a legal number for M, you should end up with the correct answer  (and  only  the correct answer).

An easy number to pick for M here is 26 (by simply adding 16 and 10). If 26 is divided by 16, the remainder is in fact 10.

Second, note that the 4 terms don't matter individually. They give you an average, and then they're all increased by 100%.

Since all 4 terms are doubled, you can also double the average.  Thus, the new average is 52.

When you divide 52 by 16, you get 3 remainder 4.  The answer is B.

Note that other numbers for M will also work as long as they give you a remainder of 10 when divided by 16. Another easy number to pick is 10, since 10 divided by 16 results in 0 remainder 10. Then double it to find the new average, 20. And when you divide 20 by 16, you get 1 remainder 4.

Always look to make your job easier. You don't get extra points for doing things the long way.  Here, picking a number and applying the doubling directly to the average make this problem much easier than it first appears.

Rafi9brokler · Answer

The arithmetic mean (average) M of 4 terms is an integer. When M is divided by 16, the remainder is 10. If each of the terms is increased by 100%, what is the remainder when the new mean is divided by 16?
here each of the terms is increased by 100%. though we do not know the number but we know the remainder and devisor. the remainder is 10 if we make it double, it becomes 20. 20 cannot be remainder because our divisor is 16. So minus 16 from 20 is 4.
so the right ans is B

Genius001 · Answer

The key observation is that increasing each term by 100% simply doubles it. Since every term is multiplied by the same factor, the arithmetic mean is also doubled.

We're given that the original mean leaves a remainder of 10 when divided by 16, so:

M ≡ 10 (mod 16)

The new mean is 2M, so:

2M ≡ 2 × 10 = 20 ≡ 4 (mod 16)

Therefore, the new mean leaves a remainder of 4 when divided by 16.

Answer: B

dhart · Answer

when all the numbers are doubled the the sum of numbers will be 4m*2 or 8m .
new avg = 2m when m divided by 16 leaves 10 as remainder , so doubling the number should also double the remainded , hence 10*2 dvided by 16 will give 4 as rem.

ans B

pappal · Answer

M=16k+10
if all terms are increased by 100% then certainly mean is doubled
2M=32k+20 when divided by 16 will get 4 as remainder.
B

SageGMATtutor · Answer

The arithmetic mean (average) M of 4 terms is an integer. When M is divided by 16, the remainder is 10. If each of the terms is increased by 100%, what is the remainder when the new mean is divided by 16?

A. 1
B. 4
C. 10
D. 12
E. 14

A. 1
B. 4
C. 10
D. 12
E. 14

This is a confusing problem, and it's hard even to know where to start. Trying out an example is a great way to understand a situation like this, and it often leads to an adequate strategy too.

"When M is divided by 16, the remainder is 10." Let's take M=26. As for the 4 terms, the simplest example is to think if they are all the same number, i.e. 16, 16, 16, 16. Next, if each of the terms is increased by 100% (i.e. doubled), then the new terms are all 52, and the average is also 52. When 52 is divided by 16, the remainder is 4 (it may help to think of the 16 times table here: 16, 32, 48, ...). So the answer is 4 (B)

If you're thinking: is one example enough? What if a different example gave a different remainder? then think what that would mean. If different examples gave different remainders, then there would be more than one possible answer and the problem would be incorrectly written. The problem implies that  all  cases will give a remainder of 4.

Also, note the deliberate over-complication here. The fact that M is the mean of 4 numbers isn't really significant: it's just a distraction that we need to work through; be ready for this kind of obfuscation in other problems.

The arithmetic mean (average) M of 4 terms is an integer.

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The arithmetic mean (average) M of 4 terms is an integer.

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615 to 715 in 15 Days (Ria's Strategies)

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